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Properties

Label 2-4000-4.3-c0-0-4
Degree $2$
Conductor $4000$
Sign $0.707 + 0.707i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618i·3-s + i·7-s + 0.618·9-s − i·11-s − 0.618·13-s + 17-s − 0.618i·19-s + 0.618·21-s − i·27-s − 29-s − 1.61i·31-s − 0.618·33-s + 0.381i·39-s + 41-s + i·43-s + ⋯
L(s)  = 1  − 0.618i·3-s + i·7-s + 0.618·9-s − i·11-s − 0.618·13-s + 17-s − 0.618i·19-s + 0.618·21-s − i·27-s − 29-s − 1.61i·31-s − 0.618·33-s + 0.381i·39-s + 41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.374371785\)
\(L(\frac12)\) \(\approx\) \(1.374371785\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 0.618iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + 0.618iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + 1.61iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - 1.61iT - T^{2} \)
53 \( 1 - 1.61T + T^{2} \)
59 \( 1 - 1.61iT - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 1.61iT - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.430274207847779664526863175322, −7.70251666556421011493892140808, −7.26788724368435078798971994073, −6.10183860542466932352451873374, −5.84748484934650119032156153366, −4.86496455404262547948516195391, −3.91013805949457686294968380816, −2.84713138838916592129009212906, −2.14938561076203802662332059353, −0.913834617035247863235072938688, 1.21181992985518142857279014885, 2.29546058780682695234630793831, 3.76310394350284417977993187261, 3.89576911498531040852320819679, 5.01081104433350568203960344771, 5.44259742859257312672396525104, 6.87297475097898418714407608611, 7.15166277542101343753912333038, 7.84329405924658451222833025972, 8.799994547703773841990276902891

Graph of the $Z$-function along the critical line